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Fuzzy fixed point theorems for fuzzy mappings via βadmissible with applications
Journal of Uncertainty Analysis and Applications volume 2, Article number: 20 (2014)
Abstract
The object of this paper is to utilize the notion of βadmissible in the sense of Mohammadi et al. (Fixed Point Theory Appl. 2013:24, 2013) to prove a fuzzy fixed point theorem and present some corollaries. We also give illustrative examples which demonstrate the validity of hypotheses of our results and applications to fuzzy fixed points for fuzzy mappings in partially ordered metric spaces are studied.
Introduction
In real world problems, fuzzy concepts were used to modeled mathematically for the purpose of automation, like regulation, production control and many others. The concept of fuzzy set was introduced by Zadeh [1]. Several researches were conducted on the generalizations of the concept of a fuzzy set. Heilpern [2] introduced the concept of fuzzy contraction mappings which maps from an arbitrary set to a certain subfamily of fuzzy sets in a metric linear space X. He also proved the existence of a fuzzy fixed point theorem which is a generalization of Nadler’s [3] fixed point theorem for multivalued mappings.
In 2001, Estruch and Vidal [4] proved a fuzzy fixed point theorem for fuzzy contraction mappings (in the Heilperns sense) over a complete metric space. Afterward, many mathematicians [5][9] generalized the result given in [4]. Recently, Abbas and Turkoglu [10] established the existence of a fuzzy fixed point theorem for fuzzy mapping of a generalized contractive mapping on a complete metric space, which generalized and extended the results in [4] and many known results in [2],[6].
On the other hand, the concept of an βadmissible mapping was introduced by Samet et al. [11]. They also give the existence of fixed point theorems via this concept. Afterward, Asl et al. [12] extended the concept of βadmissible for singlevalued mappings to multivalued mappings. Recently, Mohammadi et al. [13] introduced the concept of βadmissible for multivalued mappings which is different from the notion of β_{*}admissible which has been provided in [12]. Recently, Azam et al. [14], obtained common fixed point theorems for Chatterjea fuzzy mappings on closed balls in a complete metric space. The investigation is based on the fact that fuzzy fixed point results can be obtained simply from the fixed point theory of mappings on closed balls.
In this paper, we prove a fuzzy fixed point for a generalized contractive fuzzy mapping on a complete metric space by using the concept of βadmissible mapping in sense of Mohammadi et al. [13]. We also give illustrative example which demonstrate the validity of hypotheses of our results. As an application, we deduce fixed point theorems in partially ordered metric spaces.
Preliminaries
In this section, we recall some basic definitions and preliminaries that will be needed in this paper.
Definition 1 (1).
Let X be a space of points with generic elements of X denoted by x and I= [ 0,1]. A fuzzy subset of X is characterized by a membership function such that each element in X is associated with a real number in the interval I.
Definition 2 (2).
Let (X, d) be a metric space and A be a fuzzy set in X. If X is endowed with topology, for α∊ [0,1], the αlevel set of A, denoted by [ A]_{ α }, is defined as
where α ∊ (0,1] and for α=0 we have
where $\stackrel{}{B}$ denotes the closure of the nonfuzzy set B.
Definition 3 (15).
Let X be a nonempty set. For x ∊ X, we write {x} the characteristic function of the ordinary subset {x} of X. For α ∊ (0,1], the fuzzy point x_{ α } of X is the fuzzy set of X given by
In the sequel, I= [ 0,1], (X, d) and I^{X} denote the metric space and the collection of all fuzzy subsets of X. For A, B∊I^{X}, a fuzzy set A is said to be more accurate than a fuzzy set B (denoted by A ⊂ B) if and only if A x ≤ B x for each x in X, where A(x) and B(x) denote the membership function of A and B, respectively. For x ∊ X, S⊂X, A, B∊I^{X} and α ∊ [ 0,1], we define
and
It is easy to see that H is the Hausdorff metric on W_{ α }(X) induced by the metric d.
Definition 4 (4).
Let (X, d) be a metric space. T:X→I^{X} and α ∊ [ 0,1]. A fuzzy point x_{ α } in X is called fuzzy fixed point of T if x_{ α }⊂T x. If {x}αT x, then x is called fixed point of T.
Definition 5 (12).
Let X be a nonempty set, T:X→ 2^{X}, where 2^{X} is a collection of nonempty subsets of X and β:X×X → [0, ∝). We say that T is β_{*}admissible if
where
Definition 6 (13).
Let X be a nonempty set, T:X → 2^{X}, where 2^{X} is a collection of nonempty subsets of X and β:X×X → [0, ∞). We say that T is β_{*}admissible whenever for each x ∊X and y→T x with β(x, y) ≥ 1, we have β(y, z) ≥ 1 for all z ∊ T y.
Remark 1.
If T is β^{*}admissible mapping, then T is also βadmissible mapping.
Example 1.
Let $X=\mathbb{R}$ be endowed with the usual metric d. Define mappings T:X → 2^{X} and β:X × X → [0, ∞) by
and
Then T is βadmissible.
Fuzzy fixed point theorem
In what follows, we give some well known lemma which is useful for the proof.
Lemma 1 (2).
Let (X, d) be a metric space, x ∊ X and A ∊ I^{X}. For α ∊ [ 0,1], if p_{ α }(x, A)=0 and [ A]_{ α } is closed subset of X, then x_{ α }⊂A.
Now, we introduce the concept of βadmissible in sense of Mohammadi et al. [13] for fuzzy mapping.
Definition 7.
Let (X, d) be a metric space, β:X × X → [0, ∞), α ∊ [ 0,1] and T:X → W_{ α }(X). A mapping T is said to be βadmissible if for each x∈X and y∈ [ T x]_{ α }, with β(x, y)≥1, we have β(y, z)≥1 for all z∈ [ T y]_{ α }.
Next, we prove the existence of fuzzy fixed point theorem for some generalized type of contraction fuzzy mapping. For our result, we let Ѱ be the family of nondecreasing functions Ѱ:[0, ∞)→[0, ∞) such that $\underset{n=1}{\overset{\infty}{\Sigma}}{\psi}^{n}\left(t\right)<\infty $ for each t>0. It is easy to see that for ψ∈Ѱ, ψ(t)<t for all t>0 and ψ(0)=0.
Theorem 1.
Let (X, d) be a complete metric space, α ∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
for all x, y∈X, where L≥ 0 and
If the following condition holds,

(i)
T is βadmissible,

(ii)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that β(x _{0}, x _{1})≥1,

(iii)
if {x _{ n }} is sequence in X such that β(x _{ n }, x _{n+1})≥1 and x _{ n }→u as n→∞, then β(x _{ n }, u)≥1,

(iv)
ψ is continuous,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Proof.
For x_{0}∈X and x_{1}∈ [ T x_{0}]_{ α } in condition (ii), we get β(x_{0}, x_{1})≥1. Since [T x_{1}]_{ α } is nonempty compact subset of X, there exists x_{2}∈ [ T x_{1}]_{ α }, such that
From (2) and the fact that β(x_{0}, x_{1})≥1, we have
By the same argument, for x_{2}∈X, we have [ T x_{2}]_{ α }, which is a nonempty compact subset of X and then there exists x_{3}∈ [ T x_{2}]_{ α } such that
For x_{0}∈X and x_{1}∈ [ T x_{0}]_{ α } with β(x_{0}, x_{1})≥1, by definition of βadmissible, we get
From (1), (3) and (4), we have
By induction, we can construct a sequence {x_{ n }} in X such that, for each $n\in \mathbb{N}$, x_{ n }∈ [ T x_{n−1}]_{ α } with β(x_{n−1}, x_{ n })≥1 and
where
Hence,
for all $n\in \mathbb{N}$. If there exists ${n}^{*}\in \mathbb{N}$ which ${p}_{\alpha}({x}_{{n}^{*}},{\mathit{\text{Tx}}}_{{n}^{*}})=0$, then from Lemma 1, we have ${\left({x}_{{n}^{*}}\right)}_{\alpha}\subset {\mathit{\text{Tx}}}_{{n}^{*}}$, that is ${\left({x}_{{n}^{*}}\right)}_{\alpha}$ is a fuzzy fixed point of T. Therefore, we suppose that for each $n\in \mathbb{N}$, p_{ α }(x_{ n }, T x_{ n })>0 and thus d(x_{n−1}, x_{ n })>0 for all $n\in \mathbb{N}$. So, if d(x_{ n }, x_{n+1})>d(x_{n−1}, x_{ n }) for some $n\in \mathbb{N}$, then from (5) and ψ(t)<t for t∈(0, ∞), we have
which is a contradiction. Therefore, we have
Next, we will show that {x_{ n }} is a Cauchy sequence in X. Since continuous function ψ is belong to Ѱ, there exist ε>0 and positive integer h=h(ε) such that
Let m>n>h. Using the triangular inequality, previous relation and (6), we have
This implies that {x_{ n }} is a Cauchy sequence in X. By completeness of X, there exists x∈X such that x_{ n }→x as n→∞.
Finally, we show that p_{ α }(x, T x)=0. Assume on the contrary that p_{ α }(x, T x)>0. By condition (iii), we have β(x_{ n }, x)≥1 for all $n\in \mathbb{N}$. Now we have
Letting n→∞, it follows that
which is a contradiction. Therefore, we have p_{ α }(x, T x)=0. Hence, by Lemma 1, x_{ α }⊂T x. This complete the proof.
Next, we give some examples to support the validity of our result.
Example 2.
Let X= [ 0,1] and d:X×X→[0, ∞) as d(x, y)=x−y for all x, y∈X. Then (X, d) is a complete metric space. Let us define T:X→I^{X} by
Let $\alpha =\frac{3}{4}$. We observe that
for all x∈X. Therefore, T is fuzzy mapping from X to W_{ α }(X).
Define β:X×X→[0, ∞) by β(x, y)=2 for all x, y∈X. Then it is easy to check that T is an βadmissible. For each x, y∈X, we get
where $\psi \left(t\right)=\frac{t}{2}$ for all t>0 and L≥0. It is easy to see that conditions (ii) and (iii) in Theorem 1 hold. Therefore, all conditions of Theorem 1 hold. Thus, T has an αfuzzy fixed point x∈X, that is, a point $x=\frac{1}{6}$.
By using Remark 1, we get the following result.
Theorem 2.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
for all x, y∈X, where L≥0 and
If the following condition holds,

(i)
T is β ^{*}admissible,

(ii)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that β(x _{0}, x _{1})≥1,

(iii)
if {x _{ n }} is sequence in X such that β(x _{ n }, x _{n+1})≥1 and x _{ n }→u as n→∞, then β(x _{ n }, u)≥1,

(iv)
ψ is continuous,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
In Theorems 1 and 2, we take ψ(t)=θ t, where θ∈(0,1) then we have the following corollary which is a fuzzy extension of fixed point theorem given by Berinde [16].
Corollary 1.
Let (X, d) be a complete metric space, α∈ [0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exists β:X×X→[0, ∞) such that
for all x, y∈X, where, θ∈(0,1), L≥0 and
If the following condition holds,

(i)
T is βadmissible (or β ^{?}admissible),

(ii)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that β(x _{0}, x _{1})≥1,

(iii)
if {x _{ n }} is sequence in X such that β(x _{ n }, x _{n+1})≥1 and x _{ n }→u as n→∞, then β(x _{ n }, u)≥1,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
If we take L=0 in Theorems 1 and 2 and Corollary 1, then we have the following corollaries:
Corollary 2.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exist ψ∈Ѱ and β:X×X→[0, ∞) such that
for all x, y∈X and
If the following condition holds,

(i)
T is βadmissible (or β ^{*}admissible),

(ii)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that β(x _{0}, x _{1})≥1,

(iii)
if {x _{ n }} is sequence in X such that β(x _{ n }, x _{n+1})≥1 and x _{ n }→u as n→∞, then β(x _{ n }, u)≥1,

(iv)
ψ is continuous,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Corollary 3.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exists β:X×X→[0, ∞) such that
for all x, y∈X, where θ∈(0,1). If the following condition holds,

(i)
T is βadmissible (or β ^{*}admissible),

(ii)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that β(x _{0}, x _{1})≥1,

(iii)
if {x _{ n }} is sequence in X such that β(x _{ n }, x _{n+1})≥1 and x _{ n }→u as n→∞, then β(x _{ n }, u)≥1,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
If we set β(x, y)=1 for all x, y∈X in Theorem 1 or Theorem 2, we get the following result:
Corollary 4.
Let (X, d) be a complete metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exist ψ∈Ѱ such that
for all x, y∈X, where L≥0 and
Then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
If ψ(t)=θ t, where θ∈(0,1) and β(x, y)=1 for all x, y∈X in Theorem 1 or Theorem 2, then we have the following corollary.
Corollary 5.
Let (X, d) be a complete metric linear space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X) such that
for all x, y∈X, where θ∈(0,1). Then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Applications
In this section, we give the existence of fuzzy fixed point theorems in a partially ordered metric space.
Before presenting our results, we give the following definitions.
Definition 8.
Let X be a nonempty set. Then (X, d,⪯) is said to be an ordered metric spaces if (X, d) is a metric space and (X,⪯) is a partially ordered set.
Definition 9.
Let (X,⪯) be a partially ordered set. Then x, y∈X are said to be comparable if x⪯y or y⪯x holds.
For partially ordered set (X,⪯), we define
Definition 10.
A partially ordered set (X,⪯) is said to satisfy the order sequential limit property if $({x}_{n},x)\in \stackrel{\u2015}{\wedge}$ for all $n\in \mathbb{N}$, whenever a sequence x_{ n }→x as n→∞ and $({x}_{n},{x}_{n+1})\in \u22bc$ for all $n\in \mathbb{N}$.
Example 3.
Let $X=\mathbb{R}$ and ⪯=≤ be a partial ordered on X. It is easy to check that (X, d,⪯) satisfy the order sequential limit property.
Definition 11.
Let (X,⪯) be a partially ordered set and α∈ [ 0,1]. A fuzzy mapping T:X→W_{ α }(X) is said to be comparative if for each x∈X and y∈ [ T x]_{ α }, with $(x,y)\u22bc$, we have $(y,z)\u22bc$ for all z∈ [ T y]_{ α }.
Here we give the existence of fuzzy fixed point theorem for fuzzy mapping in ordered metric spaces.
Theorem 3.
Let (X, d,⪯) be a complete partially ordered metric space, α∈[0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exists ψ∈Ѱ such that
for all $(x,y)\u22bc$, where L≥0 and
If the following condition holds,

(I)
T is comparative fuzzy mapping,

(II)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that $({x}_{0},{x}_{1})\u22bc$,

(III)
X satisfies the order sequential limit property,

(IV)
ψ is continuous,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Proof.
Consider the mapping β:X×X→ [0, ∝) defined by
From condition (II), we get β(x_{0}, x_{1})≥1 and then the condition (ii) in Theorem 1 holds. It follows from T is comparative fuzzy mapping that T is βadmissible mapping. By (18), we have, for all x, y∈X,
Since X satisfies the order sequential limit property, the condition (iii) in Theorem 1 is also holds. Now all the hypotheses of Theorem 1 are satisfied and thus the existence of the fuzzy fixed point of fuzzy mapping T follows from Theorem 1.
By using the same technique in the proof of Theorem 3 with Corollaries 1, 2 and 3, we get the following results.
Corollary 6.
Let (X, d,⪯) be a complete partially ordered metric space, α∈[0,1] and T be fuzzy mapping from X to W_{ α }(X) such that
for all $(x,y)\in \u22bc$, where L≥0 and
If the following condition holds,

(I)
T is comparative fuzzy mapping,

(II)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that $({x}_{0},{x}_{1})\in \u22bc$,

(III)
X satisfies the order sequential limit property,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Corollary 7.
Let (X, d,⪯) be a complete partially ordered metric space, α∈[0,1] and T be fuzzy mapping from X to W_{ α }(X). Suppose that there exists ψ∈Ѱ such that
for all $(x,y)\in \u22bc$, where L≥0 and
If the following condition holds,

(I)
T is comparative fuzzy mapping,

(II)
there exist x _{0}∈X and x _{1}∈ [ T x _{0}]_{ α } such that $({x}_{0},{x}_{1})\in \u22bc$,

(III)
X satisfies the order sequential limit property,

(IV)
ψ is continuous,
then there exists x∈X such that x_{ α } is a fuzzy fixed point of T.
Corollary 8.
Let (X, d,⪯) be a complete partially ordered metric space, α∈ [ 0,1] and T be fuzzy mapping from X to W_{ α }(X) such that
for all $(x,y)\in \u22bc$, where
If the following condition holds,

(I)
T is comparative fuzzy mapping,

(II)
there exist x _{0}∊X and x _{1}∊ [ T x _{0}]_{ α } such that $({x}_{0},{x}_{1})\u220a\u22bc$,

(III)
X satisfies the order sequential limit property,
then there exists x∊X such that x_{ α } is a fuzzy fixed point of T.
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Acknowledgements
The second authors gratefully acknowledge the financial support provided by Thammasat University under the TU Research Scholar, Contract No. 2/10/2557.
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Keywords
 Fuzzy set
 Fuzzy mapping
 Fuzzy fixed point
 βadmissible mappings